// This file is part of Eigen, a lightweight C++ template library
// for linear algebra.
//
// Copyright (C) 2011 Gael Guennebaud <gael.guennebaud@inria.fr>
// Copyright (C) 2012, 2014 Kolja Brix <brix@igpm.rwth-aaachen.de>
//
// This Source Code Form is subject to the terms of the Mozilla
// Public License v. 2.0. If a copy of the MPL was not distributed
// with this file, You can obtain one at http://mozilla.org/MPL/2.0/.

#ifndef EIGEN_GMRES_H
#define EIGEN_GMRES_H

namespace Eigen {

namespace internal {

/**
 * Generalized Minimal Residual Algorithm based on the
 * Arnoldi algorithm implemented with Householder reflections.
 *
 * Parameters:
 *  \param mat       matrix of linear system of equations
 *  \param rhs       right hand side vector of linear system of equations
 *  \param x         on input: initial guess, on output: solution
 *  \param precond   preconditioner used
 *  \param iters     on input: maximum number of iterations to perform
 *                   on output: number of iterations performed
 *  \param restart   number of iterations for a restart
 *  \param tol_error on input: relative residual tolerance
 *                   on output: residuum achieved
 *
 * \sa IterativeMethods::bicgstab()
 *
 *
 * For references, please see:
 *
 * Saad, Y. and Schultz, M. H.
 * GMRES: A Generalized Minimal Residual Algorithm for Solving Nonsymmetric Linear Systems.
 * SIAM J.Sci.Stat.Comp. 7, 1986, pp. 856 - 869.
 *
 * Saad, Y.
 * Iterative Methods for Sparse Linear Systems.
 * Society for Industrial and Applied Mathematics, Philadelphia, 2003.
 *
 * Walker, H. F.
 * Implementations of the GMRES method.
 * Comput.Phys.Comm. 53, 1989, pp. 311 - 320.
 *
 * Walker, H. F.
 * Implementation of the GMRES Method using Householder Transformations.
 * SIAM J.Sci.Stat.Comp. 9, 1988, pp. 152 - 163.
 *
 */
template<typename MatrixType, typename Rhs, typename Dest, typename Preconditioner>
bool
gmres(const MatrixType& mat,
	  const Rhs& rhs,
	  Dest& x,
	  const Preconditioner& precond,
	  Index& iters,
	  const Index& restart,
	  typename Dest::RealScalar& tol_error)
{

	using std::abs;
	using std::sqrt;

	typedef typename Dest::RealScalar RealScalar;
	typedef typename Dest::Scalar Scalar;
	typedef Matrix<Scalar, Dynamic, 1> VectorType;
	typedef Matrix<Scalar, Dynamic, Dynamic, ColMajor> FMatrixType;

	const RealScalar considerAsZero = (std::numeric_limits<RealScalar>::min)();

	if (rhs.norm() <= considerAsZero) {
		x.setZero();
		tol_error = 0;
		return true;
	}

	RealScalar tol = tol_error;
	const Index maxIters = iters;
	iters = 0;

	const Index m = mat.rows();

	// residual and preconditioned residual
	VectorType p0 = rhs - mat * x;
	VectorType r0 = precond.solve(p0);

	const RealScalar r0Norm = r0.norm();

	// is initial guess already good enough?
	if (r0Norm == 0) {
		tol_error = 0;
		return true;
	}

	// storage for Hessenberg matrix and Householder data
	FMatrixType H = FMatrixType::Zero(m, restart + 1);
	VectorType w = VectorType::Zero(restart + 1);
	VectorType tau = VectorType::Zero(restart + 1);

	// storage for Jacobi rotations
	std::vector<JacobiRotation<Scalar>> G(restart);

	// storage for temporaries
	VectorType t(m), v(m), workspace(m), x_new(m);

	// generate first Householder vector
	Ref<VectorType> H0_tail = H.col(0).tail(m - 1);
	RealScalar beta;
	r0.makeHouseholder(H0_tail, tau.coeffRef(0), beta);
	w(0) = Scalar(beta);

	for (Index k = 1; k <= restart; ++k) {
		++iters;

		v = VectorType::Unit(m, k - 1);

		// apply Householder reflections H_{1} ... H_{k-1} to v
		// TODO: use a HouseholderSequence
		for (Index i = k - 1; i >= 0; --i) {
			v.tail(m - i).applyHouseholderOnTheLeft(H.col(i).tail(m - i - 1), tau.coeffRef(i), workspace.data());
		}

		// apply matrix M to v:  v = mat * v;
		t.noalias() = mat * v;
		v = precond.solve(t);

		// apply Householder reflections H_{k-1} ... H_{1} to v
		// TODO: use a HouseholderSequence
		for (Index i = 0; i < k; ++i) {
			v.tail(m - i).applyHouseholderOnTheLeft(H.col(i).tail(m - i - 1), tau.coeffRef(i), workspace.data());
		}

		if (v.tail(m - k).norm() != 0.0) {
			if (k <= restart) {
				// generate new Householder vector
				Ref<VectorType> Hk_tail = H.col(k).tail(m - k - 1);
				v.tail(m - k).makeHouseholder(Hk_tail, tau.coeffRef(k), beta);

				// apply Householder reflection H_{k} to v
				v.tail(m - k).applyHouseholderOnTheLeft(Hk_tail, tau.coeffRef(k), workspace.data());
			}
		}

		if (k > 1) {
			for (Index i = 0; i < k - 1; ++i) {
				// apply old Givens rotations to v
				v.applyOnTheLeft(i, i + 1, G[i].adjoint());
			}
		}

		if (k < m && v(k) != (Scalar)0) {
			// determine next Givens rotation
			G[k - 1].makeGivens(v(k - 1), v(k));

			// apply Givens rotation to v and w
			v.applyOnTheLeft(k - 1, k, G[k - 1].adjoint());
			w.applyOnTheLeft(k - 1, k, G[k - 1].adjoint());
		}

		// insert coefficients into upper matrix triangle
		H.col(k - 1).head(k) = v.head(k);

		tol_error = abs(w(k)) / r0Norm;
		bool stop = (k == m || tol_error < tol || iters == maxIters);

		if (stop || k == restart) {
			// solve upper triangular system
			Ref<VectorType> y = w.head(k);
			H.topLeftCorner(k, k).template triangularView<Upper>().solveInPlace(y);

			// use Horner-like scheme to calculate solution vector
			x_new.setZero();
			for (Index i = k - 1; i >= 0; --i) {
				x_new(i) += y(i);
				// apply Householder reflection H_{i} to x_new
				x_new.tail(m - i).applyHouseholderOnTheLeft(
					H.col(i).tail(m - i - 1), tau.coeffRef(i), workspace.data());
			}

			x += x_new;

			if (stop) {
				return true;
			} else {
				k = 0;

				// reset data for restart
				p0.noalias() = rhs - mat * x;
				r0 = precond.solve(p0);

				// clear Hessenberg matrix and Householder data
				H.setZero();
				w.setZero();
				tau.setZero();

				// generate first Householder vector
				r0.makeHouseholder(H0_tail, tau.coeffRef(0), beta);
				w(0) = Scalar(beta);
			}
		}
	}

	return false;
}

}

template<typename _MatrixType, typename _Preconditioner = DiagonalPreconditioner<typename _MatrixType::Scalar>>
class GMRES;

namespace internal {

template<typename _MatrixType, typename _Preconditioner>
struct traits<GMRES<_MatrixType, _Preconditioner>>
{
	typedef _MatrixType MatrixType;
	typedef _Preconditioner Preconditioner;
};

}

/** \ingroup IterativeLinearSolvers_Module
 * \brief A GMRES solver for sparse square problems
 *
 * This class allows to solve for A.x = b sparse linear problems using a generalized minimal
 * residual method. The vectors x and b can be either dense or sparse.
 *
 * \tparam _MatrixType the type of the sparse matrix A, can be a dense or a sparse matrix.
 * \tparam _Preconditioner the type of the preconditioner. Default is DiagonalPreconditioner
 *
 * The maximal number of iterations and tolerance value can be controlled via the setMaxIterations()
 * and setTolerance() methods. The defaults are the size of the problem for the maximal number of iterations
 * and NumTraits<Scalar>::epsilon() for the tolerance.
 *
 * This class can be used as the direct solver classes. Here is a typical usage example:
 * \code
 * int n = 10000;
 * VectorXd x(n), b(n);
 * SparseMatrix<double> A(n,n);
 * // fill A and b
 * GMRES<SparseMatrix<double> > solver(A);
 * x = solver.solve(b);
 * std::cout << "#iterations:     " << solver.iterations() << std::endl;
 * std::cout << "estimated error: " << solver.error()      << std::endl;
 * // update b, and solve again
 * x = solver.solve(b);
 * \endcode
 *
 * By default the iterations start with x=0 as an initial guess of the solution.
 * One can control the start using the solveWithGuess() method.
 *
 * GMRES can also be used in a matrix-free context, see the following \link MatrixfreeSolverExample example \endlink.
 *
 * \sa class SimplicialCholesky, DiagonalPreconditioner, IdentityPreconditioner
 */
template<typename _MatrixType, typename _Preconditioner>
class GMRES : public IterativeSolverBase<GMRES<_MatrixType, _Preconditioner>>
{
	typedef IterativeSolverBase<GMRES> Base;
	using Base::m_error;
	using Base::m_info;
	using Base::m_isInitialized;
	using Base::m_iterations;
	using Base::matrix;

  private:
	Index m_restart;

  public:
	using Base::_solve_impl;
	typedef _MatrixType MatrixType;
	typedef typename MatrixType::Scalar Scalar;
	typedef typename MatrixType::RealScalar RealScalar;
	typedef _Preconditioner Preconditioner;

  public:
	/** Default constructor. */
	GMRES()
		: Base()
		, m_restart(30)
	{
	}

	/** Initialize the solver with matrix \a A for further \c Ax=b solving.
	 *
	 * This constructor is a shortcut for the default constructor followed
	 * by a call to compute().
	 *
	 * \warning this class stores a reference to the matrix A as well as some
	 * precomputed values that depend on it. Therefore, if \a A is changed
	 * this class becomes invalid. Call compute() to update it with the new
	 * matrix A, or modify a copy of A.
	 */
	template<typename MatrixDerived>
	explicit GMRES(const EigenBase<MatrixDerived>& A)
		: Base(A.derived())
		, m_restart(30)
	{
	}

	~GMRES() {}

	/** Get the number of iterations after that a restart is performed.
	 */
	Index get_restart() { return m_restart; }

	/** Set the number of iterations after that a restart is performed.
	 *  \param restart   number of iterations for a restarti, default is 30.
	 */
	void set_restart(const Index restart) { m_restart = restart; }

	/** \internal */
	template<typename Rhs, typename Dest>
	void _solve_vector_with_guess_impl(const Rhs& b, Dest& x) const
	{
		m_iterations = Base::maxIterations();
		m_error = Base::m_tolerance;
		bool ret = internal::gmres(matrix(), b, x, Base::m_preconditioner, m_iterations, m_restart, m_error);
		m_info = (!ret) ? NumericalIssue : m_error <= Base::m_tolerance ? Success : NoConvergence;
	}

  protected:
};

} // end namespace Eigen

#endif // EIGEN_GMRES_H
